## Mean, Median, Mode, and Range

The mean, median, unique mode, and range of a collection of eight counting numbers are each 8. What is the largest number that can be in this collection?
Source: mathcontest.olemiss.edu 10/23/2006

SOLUTION
First solution
Let’s try a simple solution and see what we got
$1\;\;2\;\;3\;\;8\;\;8\;\;8\;\;8\;\;9$
Median = 8
Mode = 8
Range = $9-1=8$
Mean = $\frac{1+2+3+8+8+8+8+9}{8}=5.875\neq 8$
We got everything right except the mean.

Second solution
$4\;\;8\;\;8\;\;8\;\;8\;\;8\;\;8\;\;12$
Median = 8
Mode = 8
Range = $12-4=8$
Mean = $\frac{4+8+8+8+8+8+8+12}{8}=8$
We got everything right, but is 12 the largest value?

Third solution
$5\;\;6\;\;8\;\;8\;\;8\;\;8\;\;8\;\;13$
Median = 8
Mode = 8
Range = $13-5=8$

Mean = $\frac{5+6+8+8+8+8+8+13}{8}=8$
We got everything right, but is 13 the largest value?

Fourth solution
$6\;\;7\;\;7\;\;8\;\;8\;\;8\;\;8\;\;14$
Median = 8
Mode = 8
Range = $14-6=8$
Mean = $\frac{6+7+7+8+8+8+8+14}{8}=8.25\neq 8$
We got 14 as the largest value, but the mean is a little off.

Fifth solution
$6\;\;6\;\;6\;\;8\;\;8\;\;8\;\;8\;\;14$
Median = 8
Mode = 8
Range = $14-6=8$
Mean = $\frac{6+6+6+8+8+8+8+14}{8}=8$
We got everything right, but is it the end?

$7\;\;7\;\;7\;\;8\;\;8\;\;8\;\;8\;\;15$
Median = 8
Mode = 8
Range = $15-7=8$
Mean = $\frac{7+7+7+8+8+8+8+15}{8}=8.5\neq 8$
So the largest value in the collection is 14.